Otmar Scherzer
Department of Computer Science, University of Innsbruck

Tikhonov initiated the research on stable method for the numerical solution
of inverse and ill-posed problems. Tikhonov's approach consists in approximating
a solution of an operator equation

F(x)=y

by a minimizer of the penalized functional

||F(x)-y||^2 +\alpha ||x||^2 (\alpha >0) .

In the beginning mainly linear ill-posed problems (i.e. F is linear) such as computerized
tomography have been solved with these methods. The theory of Tikhonov regularization
methods developed systematically. Until around 1980 there has been success in a rigorous
and rather complete analysis of regularization methods for linear ill-posed problems.
We mention the books of Tikhonov & Arsenin, Nashed, Engl & Groetsch, Groetsch, Morozov,
Louis Natterer, Bertero & Boccacci, Kirsch, Colton & Kress... . In 1989 Engl & Kunisch &
Neubauer and Seidman & Vogel developed a regularization theory for non-linear inverse
problems where F is a non-linear, differentiable operator. About the same time Osher & Rudin
used bounded variation regularization for denoising and deblurring, which consists in
minimization of the functional

||F(x)-y||^2 + \alpha\int |\nabla x| .

This method is highly successful in restoring discontinuities. The analysis of
bounded variation regularization is significantly more involved since the
penalization functional is not differentiable. Over the past years this concept
has attracted many mathematical research. The next step toward generalization of
regularization methods is non-convex regularization. Here the general goal is to
minimize functionals of the form

\int g(F(x)-y,x,\nabla x),

which may be nonconvex with respect to the third component \nabla x.

Another complication is introduced in the analysis of regularization functionals
if for instance the operator F can be decomposed into a continuous and a discontinuous
operator. Such models have become popular for level set regularization recently.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge,
Math Annex (Room 1115).